arXiv subjects

Gunhee Cho

Publications and source records attributed to Gunhee Cho.

At least 19 recordsLinked to original sources

arXiv2026

Completing or Refusing Low-Dimensional Records of Structured Quantum Circuits: Measurement Loss, Compression Loss, and Hardware Drift

Structured quantum circuits are often summarized by low-dimensional records. Records lose control-dependent information when they merge outcomes whose probabilities respond differently to circuit parameters. We assess this loss without a parametric hardware-noise model or low-dimensional statistical family. Quantum Fisher information bounds premeasurement sensitivity; Fisher metrics of measurements and records describe device-attainable sensitivity. A Hellinger residual measures the response removed by a record, while its local quadratic term is the conditional covariance of the full-outcome score. Simultaneous confidence bounds support approval, refusal, or deferral. We prove a finite-library completion theorem: executable augmentations terminate with either a record preserving every declared response or proof that no library augmentation removes the loss. For an analytic control germ, integral closures characterize preservation along every analytic control arc, and finitely many Rees valuations detect failure. This state--measurement--record chain connects an all-arc criterion to executable completion or refusal, an attainable-Fisher local kernel criterion, and finite-sample decisions. A three-qubit calculation separates measurement loss from record loss. IBM experiments on Kingston and Marrakesh test decisions in fixed-particle-number and GHZ families, including negative controls. An end-to-end Kingston experiment reduces calibration shots by 33.3% while meeting prespecified noninferiority criteria on two held-out objectives. In two-epoch IQM Garnet data, median record-level drift is 0.0797 times full-distribution drift, but a significant residual remains in 35 of 36 settings. These experiments validate failure detection on the tested circuits; they do not establish universal compression performance or device quantum Fisher information.

quant-ph↗

arXiv2026

A Tutorial on Bregman Projection in Statistics

A single geometric operation -- projecting a reference onto a constrained family under a Bregman divergence -- underlies a striking range of statistical methods. This tutorial develops the operation first as pure convex geometry, with no statistics attached. A strictly convex generator $G$ and its conjugate $F$ furnish two coordinate systems, a projection theorem with existence and uniqueness, and a Pythagorean {theorem}; the Pythagorean theorem itself produces {two} dual projections -- the information (e-) projection onto moment-constrained families and the moment (m-) projection onto exponential families -- exchanged by the conjugacy $G\leftrightarrow F$, so a single theorem governs both. Part~II reads off the statistics. The generalized linear model is treated in detail as the concrete carrier of the two projections: {under the canonical link,} the score equation is exactly the Pythagorean orthogonality, and the fit is simultaneously an e-projection in the natural coordinate and an m-projection in the mean coordinate. Maximum entropy, survey calibration, over-identified moment models, the EM algorithm, variational inference, autoencoders, and expectation propagation then fall into place as instances of the same construction -- exactly where the underlying families are flat, and as controlled approximations or neighboring-divergence analogies where they are not. The mathematics of Part~I is self-contained; the statistical sections presume only familiarity with the methods being unified.

math.ST↗

arXiv2026

Geometry- and Topology-Informed Quantum Computing: From States to Real-Time Control with FPGA Prototypes

This book gives a geometry-first, hardware-aware route through quantum-information workflows, with one goal: connect states, circuits, and measurement to deterministic classical pipelines that make hybrid quantum systems run. Part 1 develops the backbone (essential linear algebra, the Bloch-sphere viewpoint, differential-geometric intuition, and quantum Fisher information geometry) so evolution can be read as motion on curved spaces and measurement as statistics. Part 2 reframes circuits as dataflow graphs: measurement outcomes are parsed, aggregated, and reduced to small linear-algebra updates that schedule the next pulses, highlighting why low-latency, low-jitter streaming matters. Part 3 treats multi-qubit structure and entanglement as geometry and computation, including teleportation, superdense coding, entanglement detection, and Shor's algorithm via quantum phase estimation. Part 4 focuses on topological error correction and real-time decoding (Track A): stabilizer codes, surface-code decoding as "topology -> graph -> algorithm", and Union-Find decoders down to microarchitectural/RTL constraints, with verification, fault injection, and host/control-stack integration under product metrics (bounded latency, p99 tails, fail-closed policies, observability). Optional Track C covers quantum cryptography and streaming post-processing (BB84/E91, QBER/abort rules, privacy amplification, and zero-knowledge/post-quantum themes), emphasizing FSMs, counters, and hash pipelines. Appendices provide visualization-driven iCEstick labs (switch-to-bit conditioning, fixed-point phase arithmetic, FSM sequencing, minimal control ISAs), bridging principles to implementable systems.

quant-ph↗

arXiv2026

Complex Analysis and Riemann Surfaces: A Graduate Path to Algebraic Geometry

These lecture notes present a computation driven pathway from classical complex analysis to the theory of compact Riemann surfaces and their connections to algebraic geometry. The exposition follows a compute first then abstract philosophy, in which analytic and geometric structures are introduced through explicit calculations and local models before being organized into conceptual frameworks. The notes begin with the foundations of complex analysis, including holomorphic functions, Cauchy theory, power series, residues, and contour integration, with an emphasis on hands on techniques such as Laurent expansions, residue calculus, and branch cut methods. These analytic tools are then used to construct Riemann surfaces explicitly via branched coverings and gluing constructions, which serve as recurring test cases throughout the text. Differential forms, Stokes theorem, curvature, and the Gauss Bonnet theorem provide the geometric bridge to Hodge theory, culminating in a detailed and self contained treatment of the Hodge Weyl theorem on compact Riemann surfaces, including weak formulations, regularity, and concrete examples. The algebraic geometric core develops holomorphic line bundles, divisors, the Picard group, and sheaves, followed by Cech and sheaf cohomology, the exponential sequence, and de Rham and Dolbeault theories, all treated with explicit computations. The Riemann Roch theorem is presented with full proofs and applications, leading to the construction of the Jacobian, Abel Jacobi theory, theta functions, and the correspondence between Riemann surfaces, algebraic curves, and Galois coverings. Originating from collaborative study groups associated with the Enjoying Math community, these notes are intended for graduate students seeking a concrete and unified route from complex analysis to algebraic geometry.

math.CV↗

arXiv2025

Certified Quotient Calibration with Weighted-Projective Orbits and Finite-Shot Guarantees

Periodic controls can make a quantum-calibration scan redundant, but compilation, noise, and measurement can invalidate the symmetry behind that reduction. We introduce Certified Quotient Calibration (CQC), which audits whether circuit amplitudes have a common positive grading and define weighted-projective-space (WPS) control orbits. From pilot counts it bounds the largest proposed within-orbit and smallest audited between-orbit squared Hellinger distances by U_in and L_out. CQC accepts only when these bounds meet a task tolerance, preserve audited alternatives, and imply a net saving after certification cost; otherwise it rejects or defers the reduction. The theorem bounds accepted orbit substitutions and [0,1]-valued calibration objectives on one confidence event. Its cost corollary authorizes a representative-only scan only when its certification-inclusive cost is below that of the full grid. Weighted-projective descent, Hellinger testing rates, and multinomial concentration are standard; the contribution is their combination into a finite-shot approve--reject--defer guarantee for task error and net calibration cost. Gate-level experiments on IQM Garnet and IBM Marrakesh, Kingston, and Fez exercise all three decisions in compiled one-, two-, and three-qubit families. In a preregistered two-qubit Bell-phase experiment on IBM Kingston, the pilot audit found U_in=0.01922<L_out=0.09030, and quotient and full scans selected the same correction orbit for both injected phase offsets. Fresh held-out counts met the prespecified 0.03 noninferiority margin. Including 18,432 pilot shots, the quotient procedure used 36,864 rather than 55,296 total shots, a 33.3% reduction. These results support the CQC decision path for the tested families, but not a native cubic pulse law, a device-independent WPS orbit, or a universal saving rate.

quant-ph↗

arXiv2025

Weighted Projective Line ZX Calculus: Quantized Orbifold Geometry for Quantum Compilation

We develop a unified geometric framework for quantum circuit compilation based on quantized orbifold phases and their diagrammatic semantics. Physical qubit platforms impose heterogeneous phase resolutions, anisotropic Bloch-ball contractions, and hardware-dependent $2\pi$ winding behavior. We show that these effects admit a natural description on the weighted projective line $\mathbb{P}(a,b)$, whose orbifold points encode discrete phase grids and whose monodromy captures winding accumulation under realistic noise channels. Building on this geometry, we introduce the WPL--ZX calculus, an extension of the standard ZX formalism in which each spider carries a weight--phase--winding triple $(a,\alpha,k)$. We prove soundness of LCM-based fusion and normalization rules, derive curvature predictors for phase-grid compatibility, and present the Weighted ZX Circuit Compression (WZCC) algorithm, which performs geometry-aware optimization on heterogeneous phase lattices. To connect circuit-level structure with fault-tolerant architectures, we introduce Monodromy-Aware Surface-Code Decoding (MASD), a winding-regularized modification of minimum-weight matching on syndrome graphs. MASD incorporates orbifold-weighted edge costs, producing monotone decoder-risk metrics and improved robustness across phase-quantized noise models. All results are validated through symbolic and numerical simulations, demonstrating that quantized orbifold geometry provides a coherent and hardware-relevant extension of diagrammatic quantum compilation.

quant-ph↗

arXiv2025

Structural Invariance of Green--Griffiths--Demailly Thresholds on Compact Complex Orbifolds

We prove that the Green--Griffiths--Demailly (GGD) hyperbolicity thresholds are structurally invariant. In other words, the minimal jet order and asymptotic growth rate at which invariant jet differentials appear remain unchanged when passing from a compact complex manifold to any compact smooth analytic Deligne--Mumford stack (orbifold) with the same coarse K\"ahler class. We establish an orbifold Riemann--Roch formula showing that only the identity sector contributes to the leading $m^n$ term of the Euler characteristic $\chi$, while all twisted sectors contribute only $O(m^{n-1})$. Together with curvature--positivity properties of the Demailly--Semple tower, this implies that the existence range of invariant jet differentials depends solely on the coarse K\"ahler class--hence orbifold compactification or rigidification does not alter the GGD threshold or the hyperbolicity locus.

math.AG↗

arXiv2025

Deterministic--Distance Couplings of Brownian Motions on Radially Isoparametric Manifolds

We develop a unified geometric framework for coadapted Brownian couplings on radially isoparametric manifolds (RIM)--spaces whose geodesic spheres have principal curvatures $\kappa_1(r),\dots,\kappa_{n-1}(r)$ depending only on the geodesic radius $r$. The mean curvature of such a geodesic sphere is denoted by $A(r) = \mathrm{Tr}(S_r) = \sum_{i=1}^{n-1} \kappa_i(r)$, where $S_r$ is the shape operator of the sphere of radius $r$. Within the stochastic two--point It\^{o} formalism, we derive an intrinsic drift--window inequality \[ A(r) - \sum_i |\kappa_i(r)| \;\le\; \rho'(t) \;\le\; A(r) + \sum_i |\kappa_i(r)|, \] governing the deterministic evolution of the inter--particle distance $\rho_t = d(X_t, Y_t)$ under all coadapted couplings. We prove that this bound is both necessary and sufficient for the existence of a coupling realizing any prescribed distance law $\rho(t)$, thereby extending the constant--curvature classification of Pascu--Popescu (2018) to all RIM. The endpoints of the drift window correspond to the synchronous and reflection couplings, providing geometric realizations of extremal stochastic drifts. Applications include stationary fixed--distance couplings on compact--type manifolds, linear escape laws on asymptotically hyperbolic spaces, and rigidity of rank--one symmetric geometries saturating the endpoint bounds. This establishes a direct correspondence between radial curvature data and stochastic coupling dynamics, linking Riccati comparison geometry with probabilistic coupling theory.

math.PR↗

arXiv2025

Support-Projected Petz Monotone Geometry of Pure Two-Qubit Families: Universal Three-Channel Decomposition and Non-Reduction of Curvature Invariants

We develop a support-projected Petz monotone geometry for pure two-qubit families, obtained by pulling back arbitrary Petz monotone quantum metrics to circuit-defined submanifolds and projecting onto the active spectral support of the associated quantum Fisher information tensor. This framework strictly generalizes the symmetric logarithmic derivative (SLD/Bures) case and includes, as special examples, the Wigner--Yanase and Bogoliubov--Kubo--Mori metrics among many others.

quant-ph↗

arXiv2025

Scaling Personality Control in LLMs with Big Five Scaler Prompts

We present Big5-Scaler, a prompt-based framework for conditioning large language models (LLMs) with controllable Big Five personality traits. By embedding numeric trait values into natural language prompts, our method enables fine-grained personality control without additional training. We evaluate Big5-Scaler across trait expression, dialogue generation, and human trait imitation tasks. Results show that it induces consistent and distinguishable personality traits across models, with performance varying by prompt type and scale. Our analysis highlights the effectiveness of concise prompts and lower trait intensities, providing a efficient approach for building personality-aware dialogue agents.

cs.CL↗

arXiv2025

Sharp lower bounds for the first eigenvalue of Steklov-type eigenvalue problems on a compact surface

Let $\Omega$ be a compact surface with smooth boundary and the geodesic curvature $k_g \ge {c > 0}$ along $\partial \Omega$ for some constant $c \in \mathbb{R}$. We prove that, if the Gaussian curvature satisfies $K \ge -\alpha$ for a constant $\alpha \ge 0$, then the first eigenvalue $\sigma_1$ of the Steklov-type eigenvalue problem satisfies \[ \sigma_1 + \frac{\alpha}{\sigma_1} \ge c. \] Moreover, equality holds if and only if $\Omega$ is a Euclidean disk of radius $\frac{1}{c}$ and $\alpha = 0$. Furthermore, we obtain a sharp lower bound for the first eigenvalue of the fourth-order Steklov-type eigenvalue problem on $\Omega$.

math.DG↗

arXiv2023

Probabilistic Method to Fundamental gap problems on the sphere

We provide a probabilistic proof of the fundamental gap estimate for Schr\"odinger operators in convex domains on the sphere, which extends the probabilistic proof of F. Gong, H. Li, and D. Luo for the Euclidean case. Our results further generalize the results achieved for the Laplacian by S. Seto, L. Wang, and G. Wei, as well as by C. He, G. Wei, and Qi S. Zhang. The essential ingredient in our analysis is the reflection coupling method on Riemannian manifolds.

math.PR↗

arXiv2023

Non-Measure Hyperbolicity of Complex K3 Surfaces

We show that the non-measure hyperbolicity of K3 surfaces -- which M. Green and P. Griffiths verified for certain cases in 1980 -- holds for all K3 surfaces. As a byproduct, we prove the non-measure hyperbolicity of any Hilbert schemes of points on K3 surfaces. We also obtain a new proof of the non-measure hyperbolicity of any Enriques surface.

math.AG↗

arXiv2023

Statistical Bergman geometry

This paper explores the Bergman geometry of bounded domains $\Omega$ in $\mathbb{C}^n$ through the lens of information geometry by introducing a mapping $\Phi: \Omega \rightarrow \mathcal{P}(\Omega)$, where $\mathcal{P}(\Omega)$ denotes a space of probability measures on $\Omega$. A result by J. Burbea and C. Rao establishes that the pullback of the Fisher information metric, the fundamental Riemannian pseudo-metric in information geometry, via $\Phi$ coincides with the Bergman metric of $\Omega$. Building on this idea, we consider $\Omega$ as a statistical model and present several interesting results within this framework. First, we derive a new statistical curvature formula for the Bergman metric by expressing it in terms of covariance. Second, given a proper holomorphic map $f: \Omega_1 \rightarrow \Omega_2$, we prove that if the induced measure push-forward $\kappa: \mathcal{P}(\Omega_1) \rightarrow \mathcal{P}(\Omega_2)$ preserves the Fisher information metrics, then $f$ must be a biholomorphism. Finally, we establish the consistency and the central limit theorem of the Fr\'echet sample mean for Calabi's diastasis function.

math.CV↗

arXiv2023

Sharp weighted CR trace Sobolev inequalities

We establish a sharp Sobolev trace inequality on the Siegel domain $\Omega_{n+1}$ involving the weighted norm-$W^{2,2}(\Omega_{n+1}, \rho^{1-2[\gamma]})$. The inequality is closely related the realization of fractional powers of the sub-Laplacian on the Heisenberg group $H^n=\partial \Omega_{n+1}$ as generalized Dirichlet-to-Neumann operators associated to the weighted poly-sublaplacian, generalizing observations of Frank--Gonz\'alez--Monticelli--Tan.

math.AP↗

arXiv2022

The Stochastic Schwarz lemma on K\"ahler Manifolds by Couplings and Its Applications

We first provide a stochastic formula for the Carath\'eodory distance in terms of general Markovian couplings and prove a comparison result between the Carath\'eodory distance and the complete K\"ahler metric with a negative lower curvature bound using the Kendall-Cranston coupling. This probabilistic approach gives a version of the Schwarz lemma on complete non-compact K\"ahler manifolds with a further decomposition Ricci curvature into the orthogonal Ricci curvature and the holomorphic sectional curvature, which cannot be obtained by using Yau--Royden's Schwarz lemma. We also prove coupling estimates on quaternionic K\"ahler manifolds. As a byproduct, we obtain an improved gradient estimate of positive harmonic functions on K\"ahler manifolds and quaternionic K\"ahler manifolds under lower curvature bounds.

math.DG↗

arXiv2022

Sub-Riemannian Geodesics on $SL(2, \mathbb{R})$

We explicitly describe the length minimizing geodesics for a sub-Riemannian structure of the elliptic type defined on $SL(2, \mathbb{R})$. Our method uses a symmetry reduction which translates the problem into a Riemannian problem on a two dimensional quotient space, on which projections of geodesics can be easily visualized. As a byproduct, we obtain an alternative derivation of the characterization of the cut-locus obtained in \cite{BoscaRossi}. We use classification results for three dimensional right invariant sub-Riemannian structures on Lie groups \cite{AGBD}, \cite{Biggs}, \cite{HB2} to identify exactly automorphic structures on which our results apply.

math.DG↗

arXiv2022

Vanishing results from Lichnerowicz Laplacian on complete K\"{a}hler manifolds and applications

In this paper, we show several rigidity results for harmonic $(p,q)$-forms in complete K\"{a}hler manifolds. We also give several applications to study non-compact K\"{a}hler manifolds with parallel Bochner tensor or quaternion K\"{a}hler manifolds. Our results are natural extensions of Petersen and Wink's results in \cite{PW21, PW} in the setting of complete, non-compact K\"{a}hler manifolds.

math.DG↗